# The Newton-Raphson Method

This applet investigates the Newton-Raphson iterative method for approximating roots of equations, using differentiation

###### Author and programmer: Ron Barrow

UK Year 13,KS5, GCE A Level Core Mathematics

TweetInstructions below See also: Cobweb and Staircase Diagrams

## How to Use this Applet

The function whose graph is drawn is a cubic of the form:

`y` = A`x`^{3} + B`x`^{2} +
C`x` + D, and you can vary this function by dragging the
A, B, C and D sliders at the bottom.

The Newton-Raphson Method involves the iterative formula:

`x`_{n + 1} = `x`_{n} - f(`x`_{n})/f '(`x`_{n})

which is used to try to solve the equation f(`x`) = 0.

The starting value for the iteration is represented by the light blue
circle drawn on the `x`-axis. This circle can be dragged left and
right with the mouse, to see how the iteration depends on the starting value.

You can also vary the number of iterations performed. The slider at the top right changes the number of iterations rapidly, but you can use the "Iter+" and "Iter-" buttons at the very top to change the number of iterations one at a time.

The program enables you to investigate how this iteration converges to a solution to the equation (**if** it does!), which solution is found (if any), and how quickly or slowly the convergence happens.

Play around - there are some fascinating patterns to observe!